Summary:
In this section, you will be learning to prove that two lines are parallel. Last section, you learned many theorems that cannot be used unless you know that the two lines are parallel. Finding out if two lines are parallel will follow you in your "geometrical journey" in sections with proofs. Also, knowing if two lines are parallel will help you in the events of architecture, drawing, painting, and landscaping.
Theorems: Theorem 3.8- Alternate Interior Angles Converse If two lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel
Theorem 3.9- Consecutive Interior Angles Converse If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel m<2+m<1= 180 degrees
Theorem 3.10- Alternate Exterior Angles Converse If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel
Postulates: Postulate 16- Corresponding Angles Converse If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel Examples 1. Given: m<1+m<8= 180 degrees Prove: j||k Statements Reasons
1. m<1+m<8=180 degrees 1. Given
2. <1 and <8 are supplementary 2. All angles that equal 180 degrees are supplementary
3. j||k 3. Converse of the Consecutive Interior Angles Theorem
2. Given: <6 is congruent to <8
Prove: j||k Statements Reasons
1. <6 is congruent to <8 1. Given
2. <8 is congruent to <2 2. Alternate interior angles theorem
3. j||k 3. Converse of the Alternate interior angles theorem
3. Given: <3 is congruent to <7
Prove: j||k Statements Reasons
1. <3 is congruent to <7 1. Given
2. <3 is congruent <1 2. Vertical Angles Theorem
3. <1 is congruent to <7 3. Transitive
4. <1 is congruent to <7 4. Converse of the Exterior Angles theorem
4. Given: <3 and <4 are supplementary
Prove:j||k Statements Reasons
1.<3 and <4 are supplementary 1. Given
2. <5 and <6 are supplementary 2. Linear Pair
3. <3 is congruent to <5 3. Alternate Exterior Angles theorem
4. j||k 4. Converse of the Alternate Exterior Angles Theorem
Practice Problems
1. Given: x is the transversal of lines k and j
Prove: k||j
2. Given: j is the transversal of lines x and m
Prove: x||m
3. Given: x is parallel to m
Prove: <A is supplementary to <E
Summary:
In this section, you will be learning to prove that two lines are parallel. Last section, you learned many theorems that cannot be used unless you know that the two lines are parallel. Finding out if two lines are parallel will follow you in your "geometrical journey" in sections with proofs. Also, knowing if two lines are parallel will help you in the events of architecture, drawing, painting, and landscaping.
Web Links:
http://library.thinkquest.org/20991/geo/parallel.html (Scroll down to wear it says "How to tell if lines are parallel")
http://jwilson.coe.uga.edu/emt668/EMAT6680.2001/Meyers/EMAT%206700/EMAT6700f.html
Theorems:
Theorem 3.8- Alternate Interior Angles Converse
If two lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel
Theorem 3.9- Consecutive Interior Angles Converse
If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel
Theorem 3.10- Alternate Exterior Angles Converse
If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel
Postulates:
Postulate 16- Corresponding Angles Converse
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel
Examples
1. Given: m<1+m<8= 180 degrees
Prove: j||k
Statements Reasons
1. m<1+m<8=180 degrees 1. Given
2. <1 and <8 are supplementary 2. All angles that equal 180 degrees are supplementary
3. j||k 3. Converse of the Consecutive Interior Angles Theorem
2. Given: <6 is congruent to <8
Prove: j||k
Statements Reasons
1. <6 is congruent to <8 1. Given
2. <8 is congruent to <2 2. Alternate interior angles theorem
3. j||k 3. Converse of the Alternate interior angles theorem
3. Given: <3 is congruent to <7
Prove: j||k
Statements Reasons
1. <3 is congruent to <7 1. Given
2. <3 is congruent <1 2. Vertical Angles Theorem
3. <1 is congruent to <7 3. Transitive
4. <1 is congruent to <7 4. Converse of the Exterior Angles theorem
4. Given: <3 and <4 are supplementary
Prove:j||k
Statements Reasons
1.<3 and <4 are supplementary 1. Given
2. <5 and <6 are supplementary 2. Linear Pair
3. <3 is congruent to <5 3. Alternate Exterior Angles theorem
4. j||k 4. Converse of the Alternate Exterior Angles Theorem
Practice Problems
1. Given: x is the transversal of lines k and j
Prove: k||j
2. Given: j is the transversal of lines x and m
Prove: x||m
3. Given: x is parallel to m
Prove: <A is supplementary to <E
4. Given: <A= 97 degrees <I=97 degrees
Prove: k||j